Mathematics > Dynamical Systems

Title:KAM theory and the 3D Euler equation

Abstract: We prove that the dynamical system defined by the hydrodynamical Euler
equation on any closed Riemannian 3-manifold $M$ is not mixing in the $C^k$
topology ($k > 4$ and non-integer) for any prescribed value of helicity and
sufficiently large values of energy. This can be regarded as a 3D version of
Nadirashvili's and Shnirelman's theorems showing the existence of wandering
solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the
mixing under the Euler flow of $C^k$-neighborhoods of divergence-free
vectorfields on $M$. On the way we construct a family of functionals on the
space of divergence-free $C^1$ vectorfields on the manifold, which are
integrals of motion of the 3D Euler equation. Given a vectorfield these
functionals measure the part of the manifold foliated by ergodic invariant tori
of fixed isotopy types. We use the KAM theory to establish some continuity
properties of these functionals in the $C^k$-topology. This allows one to get a
lower bound for the $C^k$-distance between a divergence-free vectorfield (in
particular, a steady solution) and a trajectory of the Euler flow.